We consider quantum mechanics on constrained surfaces which have non-Euclidean metrics and variable Gaussian curvature. The old controversy about the ambiguities involving terms in the Hamiltonian of order ${ħ}^2$ multiplying the Gaussian curvature is addressed. We set out to clarify the matter by considering constraints to be the limits of large restoring forces as the constraint coordinates deviate from their constrained values. We find additional ambiguous terms of order ${ħ}^2$ involving freedom in the constraining potentials, demonstrating that the classical constrained Hamiltonian or Lagrangian cannot uniquely specify the quantization: the ambiguity of directly quantizing a constrained system is inherently unresolvable. However, there is never any problem with a physical quantum system, which cannot have infinite constraint forces and always fluctuates around the mean constraint values. The issue is addressed from the perspectives of adiabatic approximations in quantum mechanics and Feynman path integrals, and semiclassically in terms of adiabatic actions.

The distributions of bond angles in the triatomic dihydride ion series ${\mathrm{CH}}_2^{+}$, ${\mathrm{NH}}_2^{+}$, and ${\mathrm{H}}_2{\mathrm{O}}^{+}$ have been studied using the Coulomb explosion imaging method. These distributions were measured as a function of the cooling of the internal degrees of freedom of these ions. The distribution for the coldest sample of ${\mathrm{CH}}_2^{+}$ molecules shows the most probable structure to be bent with substantial tunneling through the linear conformation. The most probable geometry for ${\mathrm{NH}}_2^{+}$ was found to be linear, though the angular distribution is significantly different from a shape of a harmonic-oscillator ground-state prediction. In the case of ${\mathrm{H}}_2{\mathrm{O}}^{+}$, we find a bent structure as expected from theory. Evidence for a linear excited state in ${\mathrm{H}}_2{\mathrm{O}}^{+}$ is seen in the hotter distributions. Comparison to the adiabatic theoretical predictions shows good agreement with the most probable geometries. However, the measured distributions are systematically wider than the squared vibronic wave functions derived from the corresponding potential-energy surfaces.

The atomic configuration interaction (CI) is reconsidered. We compare the algebraic and geometric approaches to the construction of the CI matrix and point out advantages of the latter. One-electron basis sets of quality comparable to numerical multiconfigurational Hartree-Fock are readily obtained. The generation of large CI lists of symmetry eigenfunctions is monitored by a prescription establishing an a priori identification of relevant contributions to the wave function. Systematic and well-defined truncations to multireference CI’s are examined. A formula for energy contributions of six-excited unlinked clusters is derived and shown to give reasonable estimates. Ensuing nonrelativistic CI calculations on He, Li, Be, and Ne ground states yield the lowest upper bounds in the literature, capturing 99.978, 99.923, 99.919, and 99.59% of the accepted correlation energies, respectively.

The comparison of phenomenological and supersymmetry-inspired quantum-defect methods in their relativistic and quasirelativistic formulations is undertaken through a systematic study. Derivations, which emphasize similarities and differences between these approaches, are applied to evaluate oscillator strengths for low-lying and Rydberg states for lithiumlike, sodiumlike, and copperlike ions. Relativistic transition matrix elements calculated with the Dirac wave functions are gauge invariant. For the quasirelativistic formalism, an effective transition operator in the velocity gauge is proposed for the calculations. The detailed numerical results we obtained enable us to draw inferences as to the improvements and also to the limitations of these models.

With the use of the density equation [Phys. Rev. A 14, 41 (1976)], second-order density matrices of atoms and molecules are calculated directly without any use of the wave function. The decoupling of the third- and fourth-order density matrices is done using the diagrammatic procedure of the Green’s-function method, including terms up to second order in electron correlation. In addition to the results given in a previous communication [Phys. Rev. Lett. 76, 1039 (1996)], we give more detailed formulations, discussions, and additional results for ${\mathrm{H}}_2\mathrm{O},$ ${\mathrm{NH}}_3,$ ${\mathrm{CH}}_4,$ HF, ${\mathrm{N}}_2,$ CO, and acetylene using double-ζ basis sets, and for ${\mathrm{CH}}_3\mathrm{OH},$ ${\mathrm{CH}}_3{\mathrm{NH}}_2,$ and ${\mathrm{C}}_2{\mathrm{H}}_6$ (staggered and eclipsed) using minimal basis sets. The present density-equation method gave energies as accurate as, and reduced density matrices (RDM’s) more accurate than the single and double excitation configuration interaction method. The present method seems to give better quality results as the system becomes large. The convergence was fairly good and the calculated second-order RDM’s almost satisfied some necessary conditions of the $N$ representability, the so-called $P,$ $Q,$ and $G$ conditions, while the first-order RDM’s were exactly $N$ representable. The variety of the molecules calculated and the quality of the calculated results show that the density-equation method can be a promising alternative to the wave-function approach in quantum mechanics.

A time-dependent response approach using a multireference coupled cluster wave function is presented for the calculation of frequency-dependent nonlinear properties for systems that have significant “nondynamical” electron correlation. The formulation is nonvariational in nature and is similar to Monkhorst’s linear response approach, but has to incorporate the time-dependent change of the cluster amplitudes and the model space coefficients. The static limit of the equation in the Fock space approach is shown to reduce to the time-independent formulation in the Fock space approach developed by one of us.

Density-functional theory for ensembles defined by fixed weights and fractional occupation numbers is derived using a Lagrange-multiplier version of the constrained-search procedure. This is related to an adiabatic connection between a generalized Kohn-Sham ensemble for noninteracting electrons and a minimum-energy ensemble including the electronic Coulomb interaction. Using this formalism, it is shown that discontinuous changes of the chemical potential of a finite electronic system result from constraints on density variations imposed by Fermi statistics. Such discontinuities do not exclude the possibility of defining an exchange-correlation energy functional that is free of derivative discontinuities for unconstrained density variations.

We derive general quantum-mechanical formulas for the photodetachment cross section of H${}^{-}$ in electric and magnetic fields with arbitrary orientation. The formulas are valid for any linear polarization and for any orientation. We have evaluated the cross-section spectra for different orientations of the electric and magnetic fields and displayed the effects of the orientations upon the oscillatory structures of the cross sections.

A theory of resonant elastic x-ray scattering (REXS) by free molecules is presented. Different scattering channels and their interference on the formation of the spectral profile are identified. It is shown that atomiclike contributions to the cross section are totally eliminated for orthogonal orientations of the polarization vectors of the incident and scattered x-ray photons. A strong deviation from a Lorentzian profile of the spectral bands for REXS resonances is predicted. Numerical applications for the C $K$ and O $K$ spectra of carbon monoxide demonstrate the strong dependence of the spectral profile on the excitation energy and the spectral distribution of the incident radiation. Two qualitatively different contributions to the spectral shape are identified in the limit of narrow-band excitation: the first follows the excitation energy, while the second is determined by the absorption resonances. The vibrational progressions are very frequency dependent, and at large detuning frequencies the REXS cross section collapses to a single resonance with the width equal to the width of the photon function. This result is shown to hold for resonant as well as for nonresonant contributions to the REXS cross section.

The energy shift due to the finite size of the pseudonucleus $(dd\mu {)}_{11}^{+}$ in the molecules $(dd\mu {)}_{11}e$ and $(dd\mu {)}_{11}\mathrm{dee}$, the subscripts indicating the first excited state with total angular momentum of one unit, is of importance in the theoretical estimation of the rate of $d$-$d$ fusion catalyzed by negative muons. The energy shift in the molecule $(dd\mu {)}_{11}e$ is calculated using perturbation theory up to second order. The finite-size shift is found to be 1.46 meV. This is significantly larger than the value of 0.7 meV for this energy shift calculated by Bakalov [Muon Catalyzed Fusion 3, 321 (1988)] by a method similar to the present method; recently found excellent agreement of theory with experimental results for the formation rate of the molecule $(dd\mu {)}_{11}\mathrm{dee}$ was based on Bakalov’s value with some modifications. The results of a direct calculation of the finite-size energy shifts in $(dd\mu {)}_{11}\mathrm{dee}$ using first-order perturbation theory are presented. The contribution from the quadrupole component of the $(dd\mu {)}_{11}$ charge distribution, which is not taken into account in the conventional scaling procedure based on the finite-size energy shifts of $(dd\mu {)}_{11}e$, is found to be of the order of 1 meV and to depend on the angular-momentum states of $(dd\mu {)}_{11}\mathrm{dee}$. Sources of uncertainty in the current theoretical estimates are also discussed.

By utilizing the long observation time provided by an ion storage ring and a laser probing technique, time-resolved studies of the $5d{}^4{F}_{7/2},$ $5d{}^2{G}_{9/2},$ $5d{}^2{G}_{7/2},$ and $5d{}^4{D}_{7/2}$ metastable levels about 12 eV above the ground state in the ${\mathrm{Xe}}^{+}$ ion have been performed. Extensive studies of possible systematic effects have been carried out in order to establish accurate values for the metastable lifetimes. Values for the $5d{}^4{F}_{7/2},$ $5d{}^2{G}_{9/2},$ and $5d{}^2{G}_{7/2}$ lifetimes ($57.2\pm 0.7,$ $44.7\pm 1.1,$ and $3.4\pm 1.0\mathrm{ms},$ respectively) are presented and compared with the results of theoretical calculations.

The suitability of the 411-nm ${}^2{S}_{1/2}{-}^2{D}_{5/2}$ transition in ${\mathrm{Yb}}^{+}$ is assessed as an optical frequency standard. The lifetime of the ${}^2{D}_{5/2}$ level has been measured and found to be 7.2±0.3 ms, which implies a natural linewidth of 22.2±0.9 Hz for the 411-nm transition. In addition to a decay to the ground state the ${}^2{D}_{5/2}$ level can decay into the highly metastable ${}^2{F}_{7/2}$ level where it is trapped. The branching ratio for decay into the ${}^2{F}_{7/2}$ level has been measured and found to be 0.83±0.03. A transition capable of rapidly depopulating the ${}^2{F}_{7/2}$ level and returning the ion to the cooling cycle has been demonstrated. This has enabled a quantum jump measurement of the 411-nm transition in ${}^{172}{\mathrm{Yb}}^{+}$ to be performed, yielding a center frequency of 729 476 869.13(0.42) MHz (1σ).

The sulfur $2p$ and carbon $1s$ core-electron excited states of the OCS molecule have been studied in high-resolution angular-resolved photoion experiments. Vibrationally resolved total ion yield spectra of this molecule have been recorded and several previously unobserved states have been revealed and tentatively assigned to the singly excited $ns,$ $np,$ and $\mathrm{nd}$ Rydberg levels. The $\mathrm{LS}$-split ${\pi }_{1/2}^{*}$ and ${\pi }_{3/2}^{*}$ states show significant bond length changes observed as very different intensity distributions of vibrational progressions of these states. Variations of the bond lengths of the Rydberg states have also been revealed by analyzing their vibrational structure. However, in the latter case the effect is found to be mainly dependent on the symmetry of the Rydberg electron orbital, while the state of the core electron plays a minor role. Asymmetric dissociation from the core-excited levels has been studied using the angular-resolved photoelectron-photoion technique. In general the derived values of the asymmetry parameters β support the proposed classification of the observed resonances. Remarkably, the differences between the theoretical and experimental values of the β parameters are large. The analysis of this discrepancy is supported by the results of separate triple coincidence measurements from which main fragmentation mechanisms are deduced. The coincidence measurements consistently indicate a bent geometry of the $\mathrm{S}{(2p}^{-1}){\pi }_{1/2}^{*}$ and $\mathrm{S}{(2p}^{-1}){\pi }_{3/2}^{*}$ states.

We investigate the Stark effect for spectral transitions between low-lying levels of atoms and ions. We find that often a strong cancellation of the shifts of the lower and upper levels occurs, resulting in a significant reduction in the shift. For accurate calculations, therefore, one has to account for the influence of high-principal-quantum-number and continuum levels. As examples, the shifts, splitting, and intensities of the components of the Li I $2s$-$2p$, Na I $3s$-$3p$, Ba II $5d$-$6p$, Ba II $6s$-$6p$, and the quadrupole Ba II $6s$-$5d$ (shifts only) transitions are calculated for electric fields up to $\approx$20 MV/cm. Comparison of the calculated shift parameters to experimental data for the Li I $2s$-$2p$, Na I $3s$-$3p$, and Ba II $6s$-$5d$ lines, available as yet only for low fields for which the effect is quadratic, showed an excellent agreement for most of these lines.

Theoretical results for the transition energies involving three Rydberg series, optical oscillator strengths (OOSs), and generalized oscillator strengths (GOSs) for valence and inner-shell electronic excitations are presented for the transitions ${}^{1}P\leftarrow X{}^1{S}_0(3s\to np,2s\to np,1s\to \mathrm{np}e{2p}_z\to ns),$ ${}^{1}S\leftarrow X{}^1{S}_0$($3s$→$\mathrm{ns}$, 2$s$ →$\mathrm{ns}$, 1$s$→$ns),$ and ${}^{1}D\leftarrow X{}^1{S}_0(3s\to 3d)$ in the magnesium atom. The influence of relaxation and correlation effects on the excitation energy and OOS and GOS values is studied. For this purpose, the target wave functions for each electronic state studied is determined independently at the Hartree-Fock and configuration-interaction methods. The first Born approximation is used in the calculation of the GOS and the properties between the nonorthogonal target wave functions are calculated with a biorthogonalization procedure.

Exact density functionals for hydrogenic atoms are discussed, continuing the original work of Gill and Pople [Phys. Rev. A 47, 2383 (1993)]. An exchange-correlation functional whose potential exactly cancels the Coulomb potential does give the exact density, although the potential, denoted $-v_J^Z\mathrm{}\left(r\right),$ differs significantly from the conventional BLYP [A. D. Becke, Phys. Rev. A 38, 3098 (1988); C. Lee, W. Wang, and R. G. Parr, Phys. Rev. B 37, 785 (1988)] potential. Raising $-v_J^Z\mathrm{}\left(r\right)\mathrm{}$ by the hardness of the system leads to very good agreement with BLYP in the energetically important regions, although asymptotically the BLYP potential vanishes rapidly while the shifted potential approaches the hardness of the system. These observations support the theory of Perdew, Parr, Levy, and Balduz [Phys. Rev. Lett. 49, 1691 (1982)] that exact exchange-correlation potentials do not vanish asymptotically. Examples of functionals that give exact hydrogenic densities and energies, with nonvanishing asymptotic potentials, are presented. The computed asymptotic potentials agree well with theoretical predictions, and arise naturally from the requirement that the functional be approximately applicable to all systems, and not just those with one electron.

We have investigated the $6s$ and $4f$ ionized states of the whole lanthanide series. It is shown that the calculated $6s$ ionization energies (IE’s) agree with experimental IE’s within an error of 0.6 eV, if we add the relativistic correction to those given by configuration-interaction (CI) calculations; $6s$ IE’s by CI calculations with the relativistic correction are in the range of 5.75 (La) to 6.29 eV (Lu); those by experiment are in 5.81 (La) to 6.89 eV (Lu). The relativistic effects reduce $4f$ IE’s by 2–7 eV, and the correlation effects increase those by 1–2 eV for the light lanthanides and by 2–3 eV for the heavy lanthanides. The $4f$ IE’s obtained by CI calculations with the relativistic correction agree with experiment, the error being of $\sim 2\mathrm{eV}$ throughout the whole lanthanide series. Brewer [J. Opt. Soc. Am. 61, 1666 (1971)] estimated or assigned $4f$ ionized states experimentally with the electronic configuration of ${4f}^{m-1}{5d}^n{6s}^2$ ($n=1\mathrm{}$ for ${\mathrm{Ce}}^{+},$ ${\mathrm{Gd}}^{+},$ and ${\mathrm{Lu}}^{+},$ and $n=0\mathrm{}$ for others). However, for some atoms such as ${\mathrm{Pr}}^{+},$ ${\mathrm{Nd}}^{+},$ ${\mathrm{Pm}}^{+},$ ${\mathrm{Sm}}^{+},$ and ${\mathrm{Gd}}^{+}$ we failed to obtain the ionized states having ${4f}^{m-1}{5d}^n{6s}^2$ as the main configuration.

The deexcitation cascade following photoionization of argon at energies above the $1s$ threshold gives rise to a very complex $L-MM$ Auger spectrum extending from 100 to 230 eV. This spectrum consists of many overlapping $L_{23}-MM$ spectra, each of which is emitted in the presence of a different configuration of spectator vacancies left by preceding transitions. A prerequisite of a line analysis, therefore, is the decomposition of the observed spectrum into these partial spectra. A resolution of the Auger cascade into its parallel branches has been achieved by measuring in coincidence with the Auger spectrum the final ionic charge, whereas partial spectra emitted sequentially in the same cascade branch have been identified via electron-electron coincidence spectroscopy. The results are in very good agreement with those of a published step-by-step calculation of the cascade in relativistic single-configuration average.

High-resolution measurements of the $K{\alpha }_{1,2}$ low-energy satellites were performed for the elements ${}_{42}$Mo, ${}_{44}$Ru, ${}_{46}$Pd, ${}_{48}$Cd, and ${}_{50}$Sn. The photoinduced x-ray spectra were measured using a high-resolution transmission-type bent-crystal spectrometer in modified DuMond slit geometry. Experimental evidence for the $K$-$\mathrm{LM}$ radiative Auger effect (RAE) in solid medium-mass atoms was found and particular groups of the $K$-$\mathrm{LM}$ RAE transitions were identified. The experimental intensity ratios $I(K$-$\mathrm{LM}$ RAE$)$ / $I\left(K{\alpha }_{1,2}\right)$ as well as the relative intensity of the $K$-$L_3{M}_{4,5}$ transition group were extracted from the measured spectra. A comparison of the experimental results with relativistic Hartree-Fock theoretical predictions from Scofield shows a good agreement. The experimental energies of the $K$-$\mathrm{LM}$ RAE edges are compared with calculated Auger transition energies.

It is shown that the $\Delta J=4\mathrm{}$ bifurcation effect recently discovered in the superdeformed bands of heavy nuclei also occurs in the rotational bands of diatomic molecules in the form of a $\Delta J=2\mathrm{}$ staggering. A possible explanation for this phenomenon assumes a perturbation between rotational levels that can be described in terms of parity-split $q$-deformed rotators interacting through a $q$-deformed, $J$-dependent interaction. The phenomenon is evidenced and the model is illustrated on the rotational bands of AgH.